Problem-solving practice 1 (AQA GCSE Maths): Revision Notes
Problem-solving practice 1
Overview
Approximately half of the questions in your Foundation GCSE exam will require you to problem-solve, reason, interpret or communicate mathematically. When you encounter a challenging or unfamiliar question, you can apply several effective strategies to help you work through the problem.
These skills are essential for success in mathematics examinations, as they test your ability to apply mathematical knowledge in new and unfamiliar contexts.
Key problem-solving strategies
Here are five essential strategies to help you tackle difficult exam questions:
- Sketch a diagram to help you visualise what's happening in the problem
- Try the problem with smaller or easier numbers to understand the pattern
- Plan your strategy before you begin calculating
- Write down any formulae you think might be useful
- Use x or n to represent unknown values
These strategies are particularly important for AO2 and AO3 assessment objectives, which test your mathematical reasoning and problem-solving skills. AO2 focuses on reasoning, interpreting and communicating mathematically, while AO3 tests your ability to solve problems within mathematics and in other contexts.
Problem 1: Coordinate geometry with rectangles
This problem involves finding coordinates using the properties of rectangles and their diagonals. Understanding the relationship between rectangle properties and coordinate geometry is crucial for solving these types of questions.
Worked Example: Finding Rectangle Coordinates
Given information:
- ABCD is a rectangle plotted on a coordinate grid
- The length AB is twice the length AD
- E is the midpoint where the diagonals of the rectangle intersect
- The coordinates of E are (30, 20)
Task: Work out one possible set of coordinates for points A, B, C and D (worth 4 marks)
Key insight: Since the width of the rectangle needs to be twice its height, the horizontal distance from A to E must be twice the vertical distance from A to E.
Solution approach:
- Let the vertical distance from E to a side =
- Then horizontal distance from E to a side =
- Choose for simple calculations
- This gives us coordinates: A(10, 10), B(50, 10), C(50, 30), D(10, 30)
Top tip: Choose simple numbers to make your calculations easier. The coordinates of the midpoint are multiples of 10, so try using multiples of 10 for the coordinates of the vertices as well.
Problem 2: Rearranging formulae in context
This problem demonstrates how to rearrange a formula to find an unknown value in a real-world context. Formula rearrangement is a fundamental skill that appears frequently in GCSE examinations.
Worked Example: Calculating Chicken Weight
Given formula:
Given information:
- Dexter calculates that his chicken will take exactly 1 hour 33 minutes to cook
- Task: Work out the weight of Dexter's chicken (worth 3 marks)
Method:
- Convert the time to minutes: 1 hour 33 minutes = minutes
- Let = weight in kg
- Substitute into formula:
- Solve: , so , therefore kg
Key considerations:
- You can represent the chicken's weight in kg using a single letter (such as w) to simplify your equation and make the rearrangement clearer
- The units in the formula are minutes, so make sure to convert 1 hour 33 minutes into minutes before substituting into the formula
Key Points to Remember:
- Use the five key strategies when facing unfamiliar problems: sketch, try simpler numbers, plan, write formulae, and use variables
- In coordinate problems, look for patterns and relationships between the given information
- When rearranging formulae, always check your units match throughout the calculation
- Choose simple numbers where possible to make calculations more manageable
- Always show your working clearly to maximise your marks, even if your final answer is incorrect